Lesson 5 Common Sense Solidify Understanding

Learning Focus

Use logarithms to solve exponential equations.

Solve systems of equations that contain exponential functions.

How can we use logarithms and algebraic reasoning to help us solve exponential equations?

Open Up the Math: Launch, Explore, Discuss

You already know that our number system is base , so each of the different place values in a number are powers of . That makes the base exponential and base logarithmic functions very important. Because the base logarithm is so commonly used, it is called the “common log,” and to make the notation a little more compact, it is generally written without the base. So, it is used like this:

, because

, because

See how the base is just assumed to be when a base isn’t written? This is the kind of concise notation that mathematicians just love! When you use the log key on your calculator, it is automatically base . (Some technology allows you to enter a different base for a logarithm also.)

Each of the sections below contains puzzles for you to solve about base exponential functions. You will be using base logarithms and your knowledge of exponents to find missing values for exponential functions in tables, graphs, and equations. As you are working, watch for strategies that will help you to solve equations that have a variable in the exponent.

Table Puzzles


Find the missing values of in the tables.




When you were trying to find when was in the first table, you were probably thinking of an equation like , even if you didn’t write it down. Here’s your chance! Write all the equations and their solutions that were used to find the missing in both tables. I gave you the first one. You’re welcome!

Table a: ,

Table b:


What strategy did you use to find the solutions to these equations when you were filling in the tables?

Graph Puzzles


The graph of is given below. Use the graph to solve the equations for and label the solutions.

a coordinate plane with a curved line representing to function 10 to the negative X power x555101010y505050100100100000


Label the solution with an on the graph.


Label the solution with a on the graph.


Label the solution with a on the graph.


Let’s look a little closer at the solutions that you obtained from the graph. Consider the equation:

Would you get the same result if you took the base logarithm of both sides of the equation? Try it here:

Keep going now by rewriting both sides, using logarithm properties.


Let’s try it again with . Start by taking the base logarithm of both sides of the equation, then write equivalent expressions and check your answer with the graph.


Why does this process give the same value as the graph?


One of the equations you wrote in the table puzzles was: .

How could you unwind this equation using basic operations and logarithms? Show your steps here.

Now you’re ready for the equation puzzles. Here we go!

Equation Puzzles

Solve each equation for .







Combo Puzzles

Choose any method to solve.




Ready for More?

Solve the equation using two methods:

Method 1: Use a base logarithm to find an exact expression for the solution.

Method 2: Use a base logarithm to find an approximate value for the solution.


Strategies for solving exponential equations and systems:


Lesson Summary

In this lesson, we found solutions to base exponential equations by using base logarithms to undo the equation. We used tables and graphs to help support our thinking about whether the solutions we find are reasonable. We solved systems of equations both by finding the intersection of the graphs and by finding the value that makes both equations true.



The graphs of and (both with restricted domains) are shown in the same coordinate plane.

a coordinate plane with the function f of x = 2 to the x power graphed between the points (-3,.125) and (3,8) and g of x = 2 to the -x power graphed between (-2,4) and (5,.031) x–5–5–5555y555101010000


Make a list of the features of the function .


Make a list of the features of .


Use long division to calculate . Show each step.

Write your answer as a quotient and a remainder.